Textbook

Physics for Scientists and Engineerswith Modern Physics

Third Edition Pearson Education Inc.

Fishbane/Gasiorowicz/Thornton

Fundamentals of Physics

Sixth Edition John Wiley & Sons. 2003 Halliday/Resnick/Walker

References

大学物理教程　　吴锡珑主编　　　　　　　高等教育出版社

Grading

Midterm exam 40

Final exam 50

Homework* 10

The homework must be completed before 8 o’clock on every Tuesday morning.

Email: zxgu@sjtu.edu.cn

Instructor GU Zhixia

Http://www.phycai.sjtu.edu.cn

Office hours 2:00---4:00 PM(Wednesday) 6:00---8:00 PM room 207 Shangyuan

Goals

• have a solid conceptual understanding of the fundamental physical laws

• know how these laws can be applied to solve many problems

• know how physics is relevant to the world around us

Chapter 19 The Effects of Magnetic Fields

•Magnetic effects from natural magnets have been known for a long time.

•The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece. .

•Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched.

•Bar magnet ... two poles: N and S Like poles repel; Unlike poles attract

•Small sliver of lodestone suspended with a string will always align itself in a north-south direction —it detects the earth’s magnetic field.

o5.11

19-1 Magnets and Magnetic Fields

Like poles repel Opposite poles attract

Forces between bar magnets

Magnet Magnetic Field Magnet

Magnetic fields can be mapped out using iron filings

Magnetic fields

Iron filings map the field of a horseshoe magnet

Magnetic field lines (defined in same way as electric field lines, direction and density)

a straight bar magnet a horseshoe magnet a current-carrying wire

The direction of the magnetic field of a magnet is from the north pole to the south pole

TABLE Some Approximate Magnetic Fields (T)

At the surface of a neutron star   108

Near a big electromagnet   1.5  

Near a small bar magnet   10-2

At Earth's surface   10-4  

In interstellar space   10-10  

Smallest value in a magnetically shielded room   10-14   

SI unit of tesla (T)

Electric Field Linesof an Electric Dipole

Magnetic Field Lines of a Bar Magnet

Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or - magnetic charge).

How can you isolate this magnetic charge?

Even an individual electron has a magnetic dipole

Suppose we break a bar magnet

Each fragment becomes a separate magnet with its own north and south poles.

Magnetic monopoles do not exist (as far as we know).

19-2 Magnetic Force on a Point Charge

Experiment result

BqF

v

qB

vθ

We know about the existence of magnetic fields by their effect on moving charges. The magnetic field exerts a force on the moving charge.

F

The Lorentz Force

The force on a charge q moving with velocity through a region of space with electric field and magnetic field is given by:

A Notation for Vectors Perpendicular to the Page

ACT Two protons each move at speed v (as shown in the diagram) in a region of space which contains a constant B field in the -z-direction. Ignore the interaction between the two protons. What is the relation between the magnitudes of the forces on the two

protons?

(a) F1 < F2 (b) F1 = F2 (c) F1 > F2 B

x

y

z

1

2

v

v

(a) F2x < 0 (b) F2x = 0 (c) F2x > 0

What is F2x, the x-component of the force on the second proton?

(a) decreases (b) increases (c) stays the same

Inside the B field, the speed of each proton:

19-3 Consequences of the Magnetic Force on a Charge

A static magnetic field does no work on a charge

Circular Motion in a Constant Magnetic Field

• Bv

qB

mR

v

B

qv

What will be the path q follows?Path will be circle

F

qB

mRT

22

v m

qBf

2

The period and frequency

The tracks left by charged particles moving through a bubble chamber in a magnetic field

The frequency of the circular motion does not depend on the speed (application: Cyclotron)

• Bv

B

//v

v

h

The path of the particle is a helix

v

The radius

The pitch(inclination)

A charged particle follows a helical path in a region where the magnetic field is constant.

An electron in a cloud chamber produced this 10-m-long spiral track.

• Magnetic Focusing B

Particle source A

When is small enough

qBm

Thv

v 2

//

Receiver A'

After a period, the particle which has small deflection angle will focus again.

ACT The drawing below shows the top view of two interconnected chambers. Each chamber has a unique magnetic field. A positively charged particle is fired into chamber 1, and observed to follow the dashed path shown in the figure.

1) What is the direction of the magnetic field in chamber 1?

a) Up b) Down c) Left d) Right e) Into page f) Out of page

2) What is the direction of the magnetic field in chamber 2?

a) Up b) Down c) Left d) Right e) Into page f) Out of page

3) Compare the magnitude of the magnetic field in chamber 1 to the magnitude of the magnetic field in chamber 2.

a) B1 > B2 b) B1 = B2 c) B1 < B2

• The Cyclotron

a top view of the region of a cyclotron in which the particles (protons) circulate.

A uniform magnetic field is directed up from the plane of the page. Circulating protons spiral outward within the hollow dees, gaining energy every time they cross the gap between the dees

m

qBfosc 2

Applications

• Velocity Selector

A particle moving in a region where there are both electric and magnetic fields can experience no net force

Only particles of a specific velocity will cross the region undeflected

• Charge-to-Mass Ratio of the Electron

First accelerate electrons through a known potential difference V, and then adjust electric and magnetic fields so they are not deflected. Thomson’s Apparatus

Thomson (1897) measures q/m ratio for “cathode rays” All have same q/m ratio, for any material source.Electrons are a fundamental constituent of all matter!

N: ion sourceN: ion source

RR

+ -

PP

NN

BB

• Mass Spectrometer

P: velocity selector

B’

Deflection depends on mass:

Lighter deflects more, heavier less

Mass spectrometry is a technique for separating ions by their mass-to-charge (m/z) ratios

*Motion of a Point Charge in a Nonuniform Magnetic Field

Magnetic field can confines a charged particle spiraled around a field line.

B R

• Magnetic Confinement

B

A charged particle spiraling in a magnetic field, which is strong at both ends and weak in the middle, the particle becomes trapped and moves back and forth spiraling around the field lines.

• “Magnetic Bottle”

The Van Allen radiation belts

Electrons and protons are trapped by the magnetic field of Earth. They form the Van Allen radiation belts

When particles strike the upper atmosphere and fluoresce, causing the polar aurora.

* The Mixing of Electric and Magnetic Fields

O

y

x

sS'

O'

y'

x'

u

S:

S’:

Since this holds whatever is, so we have

Observers in different inertial frames see different combinations of electric and magnetic fields

The general transformation formula for special relativity are

From the example we know that it is totally relative to divide electromagnetic field into certain electric field and magnetic field.

19-4 The Hall Effect (1879)

When a conducting strip carrying a current is placed in a magnetic field, the magnetic force on the charge carriers causes a separation of charge called the Hall effect.

Experiment result

d

IBKVab

d

I

B

a

b

In equilibrium situation

Analysis

vl

d

I

B

a

+q

mf

+ + + +

– – – –ef

E

E

b

• Hall effect in hybrid semiconductors is used to distinguish experimentally between n-type and p- type materials.

B

+ + + +

– – – –a

b

ba VV

a

b

+ + + +

– – – –

ba VV 0K0K

I I

v

q

p-type

v

q

B

Discussion

• Hall effect is used to measure the number density of charge carriers in conductors or semiconductors.

n-type

nqd

IBVab

High temperature gas (plasm) B

• The Magnetohydrodynamic (MHD) Generator

• Magnetic field can be measured using Hall probe

19-5 Magnetic Forces on Currents

Currents consist of moving charges, so will experience force in magnetic field.

Magnetic Forces on Infinitesimal Wires with Currents

Consider a current-carrying wire in the presence of a magnetic field .

There will be a force on each of the charges moving in the wire. What will be the total force dF on a length dl of the wire?

Suppose current is made up of n positive charges/volume each carrying charge q and moving with velocity v through a wire of cross-section A.

has the infinitesimal magnitude and the direction of the current

Magnetic Forces on Finite Wires with Currents

(3) For a straight wire of length L making an angle θ with the magnetic field:

We define the vector as having the length of the wire and the direction of the current

(1)

(2) For uniform magnetic field

In uniform magnetic field , the force acting on a current loop is

Discussion

BLIF

x

y

O A

I

L

B

For a current element lI

d

lI

dF

d

Example A wire of current I snakes along an arbitrary curved path in the x-y plane， A uniform magnetic field is perpendicular to the plane， find the total force on the wire.

Solution

The force on the wire is the same as that on a straight wire between O and A. The force is in the direction of y.

F

　 x

I

c d

Example A circular loop has radius R and carries current I in the clockwise direction. A magnetic field B exists in the negative z-direction. Find the tension in the loop.

y

Solution The force on a small segment

The net force on loop is 0

　

f

TT

Consider the upper half loop. The forces on it are f, TIt is in equilibrium.

CDF

ABF

Torque on current loops in a uniform magnetic field

B

1l

2l

DAF

BCF

D

CB

AI

the torque about the center axis

B

+n

A(B)

D(C)

19-6 Magnetic Force on Current Loops

Consider loop in magnetic fieldas on right

The net force on loop is 0. But there is a torque.

n

We can define the magnetic dipole moment of a current loop as follows:

direction:

magnitude:

Torque on loop can then be rewritten as:

Note: if loop consists of N turns

CDF

ABF

B

+n

A(B)

D(C)

Discussion

(1) The torque tends to increase the magnetic flux

stable equilibrium

(3) Current loops in non-uniform magnetic field

The loop will move and rotate.

B

+n

A(B)

D(C)

unstable equilibrium

(2) The equation also applies to non-rectangular loop

Electric Dipole Analogy

(per turn)

Bx

.

E

.

+q

-q

Bar Magnet Analogy

You can think of a magnetic dipole moment as a bar magnet:

– In a magnetic field they both experience a torque trying to line them up with the field

– We will see in the next chapter that such a current loop does produce magnetic fields, similar to a bar magnet. In fact, atomic scale current loops were once thought to completely explain magnetic materials (in some sense they still are!).

=N

S

Galvanometer

The galvanometer: a device that measures currents

The galvanometer uses the fact that a magnetic field exerts a toque on a current loop to measure currents:

– In this picture the loop (and hence the needle) experiences a torque

– The spring produces a countertorque

– The needle will sit at its equilibrium position

– A larger current means a larger torque

Galvanometers are at the heart of many ammeters and voltmeters.

Electric Motor

A current-carrying loop is placed in the magnetic field.

A split-ring commutator is used to change the direction of the current every time the loop passes 180o

The torque on the loop serves to turn the loop

Example A plastic disk with radius R1 and R2 is placed in the uniform magnetic field . The charge q is uniformly distributed on the disk. The disk rotates about its axis with .Find the magnetic dipole moment of the disk and the torque on the disk.

R1R2

Solution Consider a circular strip

B

+n

A(B)

D(C)

The work done by the magnetic force

The minus sign arises because the torque tends to decrease

Potential Energy of Dipole

The work depends only on position, we can introduce the potential energy.

Define a potential energy U with zero at position of max torque

A current loop in a magnetic field has potential energy, depending on the angle between µ and B .

B

x

Bx

B

x

= 0

U = -B

= 0

U = B

positive work negative work

=B

X

U = 0

stable unstable

ACT A circular loop of radius R carries current I as shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.

(c) It will not rotate

(1)The coil will rotate to which of the following positions?

I

R

x

y B

a b

a b

(a)x

y

(b)

a b

x

y

(a) U0 is minimum

(b) U0 is maximum

(c) neither

(2) What is the potential energy U0 of the loop in its initial position?

I

R

x

y B

a b

Example Figure below shows a circular coil with 250 turns, an area A of 2.52 * 10-4 m2, and a current of 100 A. The coil is at rest in a uniform magnetic field of magnitude B = 0.85 T, with its magnetic dipole moment   initially aligned with   .

(a)  what is the direction of the current in the coil?

(b)  How much work would the torque applied by an external agent have to do on the coil to rotate it 90° from its initial orientation, so that    is perpendicular to    and the coil is again at rest?

Solution

(a) applying the right-hand rule to the coil ,the direction of the current in the wires on the near side of the coil is from top to bottom.

(b) work done by an external agent

Example A rectangular wire loop of width a and height b is connected to a current source that, when turned on, gives rise to a current I in the wire. The loop is suspended in a uniform magnetic field that points in a vertical direction, and it would hang vertically if there were no current. We assume that the wire is massless, but two masses m are suspended at the lower corners. What is the angle at which the loop is in equilibrium? What happens if the direction of the current is reversed?

Solution

The magnetic dipole moment

At equilibrium, the net torque is zero

if the direction of the current is reversed, the loop will rise to the same angle on the other side of the vertical

pivot

b

2mg

Another way :

Use the condition of the minimal potential energy at equilibrium

Choose the vertical plane as the reference position for the potential energy

pivot

b

2mg